Three-dimensional, incompressible turbulent jets with rectangular and elliptical cross-sections are simulated with a finite-difference numerical method. The full Navier- Stokes equations are solved at low Reynolds numbers, whereas at high Reynolds numbers filtered forms of the equations are solved along with a sub-grid scale model to approximate the effects of the unresolved scales. A 2-N storage, third-order Runge-Kutta scheme is used for temporary discretization and a fourth-order compact scheme is used for spatial discretization. Although such methods are widely used in the simulation of compressible flows, the lack of an evolution equation for pressure or density presents particular difficulty in incompressible flows. The pressure-velocity coupling must be established indirectly. It is achieved, in this study, through a Poisson equation which is solved by a compact scheme of the same order of accuracy. The numerical formulation is validated and the dispersion and dissipation errors are documented by the solution of a wide range of benchmark problems. Three-dimensional computations are performed for different inlet conditions which model the naturally developing and forced jets. The experimentally observed phenomenon of axis-switching is captured in the numerical simulation, and it is confirmed through flow visualization that this is based on self-induction of the vorticity field. Statistical quantities such as mean velocity, mean pressure, two-point velocity spatial correlations and Reynolds stresses are presented. Detailed budgets of the mean momentum and Reynolds stresses are presented. Detailed budgets of the mean momentum and Reynolds stress equations are presented to aid in the turbulence modeling of complex jets. Simulations of circular jets are used to quantify the effect of the non-uniform curvature of the non-circular jets.